On asymptotic behavior of least squares estimators and the confidence intervals of the superimposed exponential signals

The problem of estimation of the parameters of complex sinusoids in complex white noise arises in many fields. Although many papers appeared in the last twenty years dealing with the estimation of the complex sinusoids, not much attention has been paid to obtain the confidence intervals of the unknown parameters, without which the estimation may not make much sense. Recently Rao and Zhao (1993) obtained the asymptotic distribution of the least squares estimators of the frequencies of the complex sinusoidal model under the assumption of the Gaussian white noise, which can be used to obtain the confidence interval of the unknown parameters for finite sample. However, they did not perform any numerical study to see the validity of the asymptotic results for finite sample sizes. Moreover, in many situations it is observed that the error distributions need not be Gaussian. In this paper we consider the superimposed exponential model when the error distributions may not be Gaussian. We prove the strong consistency of the least squares estimators and derive the asymptotic distributions of the least squares estimators, which can be used to obtain the confidence interval of the unknown parameters. Finally some numerical experiments are performed to see how the asymptotic results behave for finite sample sizes and for different error distributions. ( 1999 Elsevier Science B.V. All rights reserved.